Varying-payment annuity functions (Chapter 8)

Chapter 8 non-level annuity functions for increasing and decreasing life annuities.

Computes $$(Ia)_x = \sum_{t=1}^{\infty} t \, v^t \, {}_tp_x.$$

Computes $$(Ia)_{x:\overline{n}|} = \sum_{t=1}^{n} t \, v^t \, {}_tp_x.$$

Computes $$(Da)_{x:\overline{n}|} = \sum_{t=1}^{n} (n+1-t)\, v^t \, {}_tp_x.$$

Computes $$(I\ddot{a})_x = \sum_{t=0}^{\infty} (t+1)\, v^t \, {}_tp_x.$$

Computes $$(I\ddot{a})_{x:\overline{n}|} = \sum_{t=0}^{n-1} (t+1)\, v^t \, {}_tp_x.$$

Computes $$(D\ddot{a})_{x:\overline{n}|} = \sum_{t=0}^{n-1} (n-t)\, v^t \, {}_tp_x.$$

Computes $$(\bar{I}\bar{a})_x = \int_0^\infty t\,v^t\,{}_tp_x\,dt.$$

Computes $$(\bar{I}\bar{a})_{x:\overline{n}|} = \int_0^n t\,v^t\,{}_tp_x\,dt.$$

Computes $$(\bar{D}\bar{a})_{x:\overline{n}|} = \int_0^n (n-t)\,v^t\,{}_tp_x\,dt.$$

Usage

Iax(x, i, model, ..., k_max = 5000, tol = 1e-12)

Iaxn(x, n, i, model, ...)

Daxn(x, n, i, model, ...)

Iadotx(x, i, model, ..., k_max = 5000, tol = 1e-12)

Iadotxn(x, n, i, model, ...)

Dadotxn(x, n, i, model, ...)

Iabarx(x, i, model, ..., tol = 1e-10)

Iabarxn(x, n, i, model, ...)

Dabarxn(x, n, i, model, ...)

Arguments

x

Age.

i

Effective annual interest rate.

model

Survival model.

...

Additional model parameters.

k_max

Maximum summation horizon for non-terminating models.

tol

Truncation tolerance for non-terminating models.

n

Term in years.

Value

Numeric vector.

Numeric vector.

Numeric vector.

Details

The functions implemented here match the notation in Section 8.6:

• Iax() = \((Ia)_x\)

• Iaxn() = \((Ia)_{x:\overline{n}|}\)

• Daxn() = \((Da)_{x:\overline{n}|}\)

• Iadotx() = \((I\ddot{a})_x\)

• Iadotxn() = \((I\ddot{a})_{x:\overline{n}|}\)

• Dadotxn() = \((D\ddot{a})_{x:\overline{n}|}\)

• Iabarx() = \((\bar{I}\bar{a})_x\)

• Iabarxn() = \((\bar{I}\bar{a})_{x:\overline{n}|}\)

• Dabarxn() = \((\bar{D}\bar{a})_{x:\overline{n}|}\)